Solving Equations Graphically: 2x+3y=50, -2x+3y=30

Hey guys! Today, we're diving into solving a system of equations using the graphical method. It might sound intimidating, but trust me, it's super cool once you get the hang of it. We’ll be tackling the equations 2x + 3y = 50 and -2x + 3y = 30. Think of it like this: we're plotting these equations as lines on a graph, and the point where they intersect is our solution. Ready to jump in and see how it's done? Let's break it down step by step!

Understanding the Graphical Method

The graphical method is a fantastic way to visualize solutions to systems of equations. At its core, this method involves plotting the equations on a coordinate plane and identifying the point(s) where the lines intersect. This intersection represents the solution that satisfies all the equations in the system. For linear equations, which form straight lines, the solution is simply the point where the lines cross each other. If the lines are parallel, they don't intersect, meaning there's no solution. And if they overlap completely, there are infinitely many solutions. This visual approach not only helps in finding the solution but also gives a clear understanding of the relationship between the equations. In the equations we're tackling today, 2x + 3y = 50 and -2x + 3y = 30, we’re looking for a single point (x, y) that makes both equations true. So, let’s roll up our sleeves and graph these lines to find out where they meet!

Step 1: Rewrite the Equations in Slope-Intercept Form

Okay, so first things first, to graph these equations easily, we need to get them into what's called slope-intercept form. Remember that? It's that y = mx + b format, where 'm' is the slope and 'b' is the y-intercept. This form makes it super simple to plot the lines. Let's start with our first equation, 2x + 3y = 50. We need to isolate 'y' on one side. So, we subtract 2x from both sides, which gives us 3y = -2x + 50. Now, we divide everything by 3, and bam! We have y = (-2/3)x + (50/3). That's our first equation ready to graph. Now, let's do the same for the second equation, -2x + 3y = 30. We add 2x to both sides, giving us 3y = 2x + 30. Then, we divide by 3, and voila! We get y = (2/3)x + 10. Now we have both equations in slope-intercept form, making them much easier to graph. This step is crucial because it lays the groundwork for visually representing our equations and finding their intersection point.

Step 2: Plot the Equations on a Graph

Alright, guys, grab your graph paper (or your favorite graphing app!) because it's plotting time! We've got our equations in slope-intercept form, which makes this part a breeze. Remember, the equation y = (-2/3)x + (50/3) tells us that the line has a y-intercept of 50/3 (which is about 16.67) and a slope of -2/3. This means for every 3 units we move to the right on the x-axis, we move 2 units down on the y-axis. Plot a couple of points using this slope and y-intercept, then draw a line connecting them. That's our first line down! Next up is y = (2/3)x + 10. This line has a y-intercept of 10 and a slope of 2/3. So, for every 3 units we move right on the x-axis, we move 2 units up on the y-axis. Plot those points, connect them, and we've got our second line. The key here is accuracy – the more precise your lines, the easier it will be to pinpoint where they intersect. So, take your time, double-check your points, and get those lines plotted perfectly. This visual representation is where the magic happens, as we’ll see the solution to our system of equations right before our eyes!

Step 3: Identify the Point of Intersection

Okay, we've got our lines plotted, and now comes the exciting part – finding where they cross! The point where the two lines intersect is the solution to our system of equations. It's the one and only spot that satisfies both 2x + 3y = 50 and -2x + 3y = 30. Take a close look at your graph and see if you can clearly identify the coordinates of this intersection point. It might be a whole number, or it could be a fraction or decimal. Eyeballing it is a good start, but to be super precise, you might want to use a ruler or zoom in if you're using a graphing app. In our case, if you've graphed everything accurately, you should see the lines intersecting at the point (5, 40/3). This means that x = 5 and y = 40/3 (or about 13.33). This point is the key – it's the solution that makes both equations true! Identifying this intersection visually is what makes the graphical method so powerful and intuitive. But to be absolutely sure, let’s plug these values back into our original equations and verify.

Step 4: Verify the Solution

Alright, we've found our potential solution – x = 5 and y = 40/3. But to be absolutely sure, we need to verify it. This means plugging these values back into our original equations, 2x + 3y = 50 and -2x + 3y = 30, to see if they hold true. Let's start with the first equation. Substituting x = 5 and y = 40/3, we get 2(5) + 3(40/3) = 10 + 40 = 50. Bingo! It works for the first equation. Now, let's try the second equation. Plugging in the same values, we have -2(5) + 3(40/3) = -10 + 40 = 30. Double bingo! It works for the second equation too. This verification step is super important because it confirms that we haven't made any mistakes along the way. It's like the final seal of approval on our solution. So, now we can confidently say that (5, 40/3) is indeed the solution to our system of equations. We’ve nailed it!

Conclusion

So there you have it, guys! We've successfully solved the system of equations 2x + 3y = 50 and -2x + 3y = 30 using the graphical method. We started by rewriting the equations in slope-intercept form, then plotted them on a graph, identified the point of intersection, and finally verified our solution. The intersection point, (5, 40/3), is the magical solution that makes both equations true. The graphical method is such a powerful tool because it allows us to visualize the solution. It's not just about numbers; it's about seeing how these equations relate to each other on a coordinate plane. Whether you're tackling algebra homework or just love solving puzzles, this method is a fantastic one to have in your toolkit. Keep practicing, and you'll become a pro at solving systems of equations graphically in no time! Remember, math can be fun, especially when you can see the answers right before your eyes on a graph. Keep exploring, keep learning, and most importantly, keep having fun with math!